Im wanting to upgrade my stock turbo on my 04 WRX to a VF34, and im also running Cobb AP. The Cobb map calls for the stock intercooler with this turbo, so would it hurt or affect anything if i ran this turbo, the map for it, but with an Sti intercooler instead of my stock intercooler? Also, is there any other parts ill need besides my AP, and fuel system to run this turbo or with my stock wastegate, etc. be ok?

You dont need a JDM STI I/C, just an STI I/C would do you fine. But if you are running Cobbs OTS map and have what is in their map notes you will be fine. But if you change things a little, say injectors or a bigge TMIC you would probably be fine also, just wont get the gains from them.